Tuesday, February 27, 2018
(in North America and South America)
Wednesday, February 28, 2018
(outside of North American and South America)
©2017 University of Waterloo
When we arrange the five choices from smallest to largest, we obtain
Thus,
Answer: (B)
Evaluating,
Answer: (A)
July 14 is 11 days after July 3 of the same year.
Since there are 7 days in a week, then July 10 and July 3 occur on the same day of the week, namely Wednesday.
July 14 is 4 days after July 10, and so is a Sunday.
Answer: (C)
Since the car is charged 3 times per week for 52 weeks, it is charged
Since the cost per charge is $0.78, then the total cost is
Answer: (E)
Since
Answer: (A)
Solution 1
Consider the
Each of these rectangles is divided in half by its diagonal, and so is half shaded.
Therefore,
Solution 2
The entire
Both shaded triangles have height 6 (the height of the square).
The bases of the triangles have lengths 2 and 4:
The left-hand triangle has area
The right-hand triangle has area
Therefore, the total shaded area is
Answer: (A)
There are
When Stephen removes a tie at random, the probability of choosing a pink tie is
Answer: (C)
The section of the number line between 0 and 5 has length
Since this section is divided into 20 equal parts, the width of each part is
Since
Since
Therefore,
Answer: (E)
If
If
If
If
Thus,
This means that
Answer: (D)
The line that passes through
This means that, for every 1 unit to the right, the line moves 1 unit up.
Thus, moving 2 units to the right from the starting point
Therefore, the line passes through
Answer: (C)
The complete central angle of a circle measures
This means that the angle in the circle graph associated with Playing is
A central angle of
This means that the baby polar bear plays for
Answer: (C)
Of the given uniform numbers,
11 and 13 are prime numbers
16 is a perfect square
12, 14 and 16 are even
Since Karl’s and Liu’s numbers were prime numbers, then their numbers were 11 and 13 in some order.
Since Glenda’s number was a perfect square, then her number was 16.
Since Helga’s and Julia’s numbers were even, then their numbers were 12 and 14 in some order. (The number 16 is already taken.)
Thus, Ioana’s number is the remaining number, which is 15.
Answer: (D)
Since the given equilateral triangle has side length 10, its perimeter is
In terms of
Since the two perimeters are equal, then
Since the rectangle is
Since
Answer: (B)
The average of the numbers
The average of the numbers
We can check that the averages of the remaining three combinations of four numbers is not equal to the fifth number.
Therefore, the answer is 11.
(We note that in fact the average of the original five numbers is
Answer: (D)
We would like to find the first time after 4:56 where the digits are consecutive digits in increasing order.
It would make sense to try 5:67, but this is not a valid time.
Similarly, the time cannot start with 6, 7, 8 or 9.
No time starting with 10 or 11 starts with consecutive increasing digits.
Starting with 12, we obtain the time 12:34. This is the first such time.
We need to determine the length of time between 4:56 and 12:34.
From 4:56 to 11:56 is 7 hours, or
From 11:56 to 12:00 is 4 minutes.
From 12:00 to 12:34 is 34 minutes.
Therefore, from 4:56 to 12:34 is
Answer: (A)
First, we note that we cannot have
After 6 X’s and 3 Y’s, there are twice as many X’s as Y’s. In this case,
After 6 X’s and 12 Y’s, there are twice as many Y’s as X’s. In this case,
The next letters are all Y’s (with 24 Y’s in total), so there are no additional values of
At this point, there are 6 X’s and 24 Y’s.
After 24 Y’s and 12 X’s (that is, 6 additional X’s), there are twice as many Y’s as X’s. In this case,
After 24 Y’s and 48 X’s (that is, 42 additional X’s), there are twice as many X’s as Y’s. In this case,
Since we are told that there are four values of
Answer: (C)
We note that
Since
Finding the number of possible values of
The prime numbers that are at most 31 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.
The distinct products of pairs of these that are at most 31 are:
Therefore, there are 7 such values of
Answer: (E)
Consider
Since the sum of the angles in a triangle is
Since
Therefore,
Answer: (E)
We recall that
Travelling
We are told that the difference between these lengths of time is 16 minutes.
Since there are 60 minutes in an hour, then 16 minutes is equivalent to
Since the time at 120 km/h is 16 minutes less than the time at 90 km/h, then
Combining the fractions on the left side using a common denominator of
Thus,
Since
Answer: (D)
We drop a perpendicular from
Since quadrilateral
Next,
Since
Since
Thus, the area of
By the Pythagorean Theorem,
Since
Since
By the Pythagorean Theorem,
Therefore, the area of
Finally, the area of pentagon
Answer: (D)
We determine the number of ways to get to each unshaded square in the grid obeying the given rules.
In the first row, there is 1 way to get to each unshaded square: by starting at that square.
In each row below the first, the number of ways to get to an unshaded square equals the sum of the number of ways to get to each of the unshaded squares diagonally up and to the left and up and to the right from the given square. This is because any path passing through any unshaded square needs to come from exactly one of these unshaded squares in the row above.
In the second row, there are 2 ways to get to each unshaded square: 1 way from each of two squares in the row above.
In the third row, there are 2 ways to get to each of the outside unshaded squares and 4 ways to get to the middle unshaded square.
Continuing in this way, we obtain the following number of ways to get to each unshaded square:
Since there are 6, 12 and 6 ways to get to the unshaded squares in the bottom row, then there are
Answer: (D)
Each wire has 2 ends.
Thus, 13 788 wires have
In a Miniou circuit, there are 3 wires connected to each node.
This means that 3 wire ends arrive at each node, and so there are
Answer: (B)
The circle with centre
This means that
Therefore, the circle with diameter
To find the area of the shaded region, we calculate the area of the region common to both circles and subtract the area of the circle with diameter
Suppose that the two circles intersect at
Join
By symmetry, the area of the shaded region on each side of
The area of the shaded region on the right side of
Figure 1
Figure 2
Since each of the large circles has radius 1, then
This means that each of
Therefore,
Lastly, consider
Note that
Since
By symmetry,
By the Pythagorean Theorem in
Therefore,
The area of
Now, we can calculate the area of the shaded region to the right of
Therefore, the area of the shaded region with the circle with diameter
Answer: (E)
We refer to the students with height 1.60 m as “taller” students and to those with height 1.22 m as “shorter” students.
For the average of four consecutive heights to be greater than
If there are 2 taller and 2 shorter students, then the sum of their heights is
Therefore, there must be more taller and fewer shorter students in a given group of 4 consecutive students.
If there are 3 taller students and 1 shorter student, then the sum of their heights is equal to
Thus, in Mrs. Warner’s line-up, any group of 4 consecutive students must include at least 3 taller students and at most 1 shorter student. (4 taller and 0 shorter students also give an average height that is greater than
For the average of seven consecutive heights to be less than
Note that 6 taller students and 1 shorter student have total height
Thus, in Mrs. Warner’s line-up, any group of 7 consecutive students must include at most 5 taller students and at least 2 shorter students.
Now, we determine the maximum possible length for a line-up. We use “T” to represent a taller student and “S” to represent a shorter student.
After some fiddling, we discover the line-up TTSTTTSTT.
The line-up TTSTTTSTT has length 9 and has the property that each group of 4 consecutive students includes exactly 3 T’s and each group of 7 consecutive students includes exactly 5 Ts and so has the desired average height properties.
We claim that this is the longest such line-up, which makes the final answer 9 or (D).
Suppose that there was a line-up of length at least 10, and the first 10 heights in the line-up were
Each row in this table is a list of 7 consecutive people from the line-up
Each column in this table is a list of 4 consecutive people from the line-up
The sum of the numbers in this table equals the sum of the sums of the 4 rows, which must be less than
The sum of the numbers in this table equals the sum of the sums of the 7 columns, which must be greater than
The sum cannot be both less than and greater than 42.00 m.
This means that our assumption is incorrect, and so it must be impossible to have a line-up of length 10 or greater.
(There are a number of other ways to convince yourself that there cannot be more than 9 students in the line-up.)
Answer: (D)
Suppose that
Since
Since
Since
In other words,
But
In other words,
Since
In other words,
But
In other words,
But
Since
We know that
If
If
If
Suppose that
Since
Therefore,
Since
For each possible pair
the remainder when
We make a table:
Divisors of |
Possible |
|||
---|---|---|---|---|
5 | 10 | 490 = |
5, 10, 35, 70, 245, 490 | 35, 245 |
5 | 15 | 485 = |
5, 485 | 485 |
4 | 8 | 492 = |
4, 12, 164, 492 | 12, 164, 492 |
4 | 12 | 488 = |
4, 8, 244, 488 | 244 |
2 | 4 | 496 = |
2, 4, 8, 16, 62, 124, 248, 496 | 62 |
2 | 6 | 494 = |
2, 26, 38, 494 | 26, 38, 494 |
2 | 8 | 492 = |
2, 4, 6, 12, 82, 164, 246, 492 | 82 |
2 | 10 | 490 = |
2, 10, 14, 70, 98, 490 | None |
2 | 12 | 488 = |
2, 4, 8, 122, 244, 488 | 122 |
2 | 14 | 486 = |
2, 6, 18, 54, 162, 486 | None |
Answer: (E)